All our numbers are built from ones (0–9) and tens (10, 20, 30…). When adding:

• First add ones with ones.
• If the ones sum reaches 10 or more — regroup: write 0 (or the remainder) in the ones place and carry 1 to the tens place.
• Then add tens with tens (including the carry).

When subtracting:

• Subtract ones from ones.
• If there aren't enough ones — borrow one ten from the tens column (it becomes 10 ones), then subtract.
• Then subtract tens from tens (accounting for the borrow).

A handy tool: the number line — draw a line, mark the first number, and jump in steps of 10 and then 1.

Example 1: Simple Addition without Regrouping

Problem: Calculate: \( 34 + 25 \)

Solution:

1. Write in columns: 34 above 25.
2. Ones: \( 4 + 5 = 9 \). Write 9.
3. Tens: \( 3 + 2 = 5 \). Write 5.
4. Answer: 59.

Answer: \( 34 + 25 = 59 \)

Example 2: Addition with Regrouping

Problem: Calculate: \( 47 + 36 \)

Solution:

1. Write in columns: 47 above 36.
2. Ones: \( 7 + 6 = 13 \). Write 3 and carry 1 to the tens.
3. Tens: \( 4 + 3 + 1\text{(carry)} = 8 \). Write 8.
4. Answer: 83.

Answer: \( 47 + 36 = 83 \)

Example 3: Simple Subtraction without Borrowing

Problem: Calculate: \( 76 - 42 \)

Solution:

1. Write in columns: 76 above 42.
2. Ones: \( 6 - 2 = 4 \). Write 4.
3. Tens: \( 7 - 4 = 3 \). Write 3.
4. Answer: 34.

Answer: \( 76 - 42 = 34 \)

Example 4: Subtraction with Borrowing (Regrouping)

Problem: Calculate: \( 52 - 28 \)

Solution:

1. Write in columns: 52 above 28.
2. Ones: \( 2 - 8 \) — not enough! Borrow 10 from the tens. Now \( 12 - 8 = 4 \). Write 4.
3. Tens: we had 5 tens, borrowed one — 4 remain. \( 4 - 2 = 2 \). Write 2.
4. Answer: 24.

Answer: \( 52 - 28 = 24 \)

Example 5: Word Problem — How Many Are Left?

Problem: A classroom had 63 pencils. Children took 27 pencils home. How many pencils are left in the classroom?

Solution:

1. We need subtraction: \( 63 - 27 \).
2. Ones: \( 3 - 7 \) — too small, borrow. \( 13 - 7 = 6 \).
3. Tens: \( 6 - 1\text{(borrow)} - 2 = 3 \).
4. Answer: 36.

Answer: \( 36 \) pencils are left.

✗ Common mistake: Starting from the left (the tens) makes regrouping impossible to manage.

✓ The correct way: Always start from the right — the ones place! That way the carry or borrow flows naturally to the left.

✗ Common mistake: In subtraction with borrowing: forgetting to subtract 1 from the tens column after borrowing.

✓ The correct way: Every borrow must be paid back! After borrowing a ten, mark a small line (or a small number) on the tens digit as a reminder that it is now 1 less.

✗ Common mistake: In addition with regrouping: writing the whole number (e.g., 13) in the ones column.

✓ The correct way: In the ones column, write only the right-hand digit (3), and carry the 1 to the tens column.

• Column addition: start from the ones, work left. If sum ≥ 10 — carry 1 to the next column.
• Column subtraction: start from the ones, work left. If top digit is too small — borrow 10 from the tens.
• Always check your answer with estimation or reverse addition.